FUZZY TL-UNIFORM SPACES
KHALED A. HASHEM AND NEHAD N. MORSI
Received 12 September 2005; Revised 12 April 2006; Accepted 25 April 2006
The main purpose of this paper is to introduce a new structure that is a fuzzy TL-uniform
space. We show that our structure generates a fuzzy topological space, precisely, a fuzzy Tlocality space. Also, we deduce the concept of level uniformities of a fuzzy TL-uniformity.
We connect the category of fuzzy TL-uniform spaces with the category of uniform spaces.
We establish a necessary and sufficient condition, under which a fuzzy TL-uniformity is
probabilistic pseudometrizable. Finally, we define a functor from the category of fuzzy
TL-uniform spaces into the category of fuzzy T-locality spaces and we show that it preserves optimal lifts.
Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
1. Introduction
In this paper, we introduce, for each continuous triangular norm T, a new structure that
is a fuzzy TL-uniform space, which generates a fuzzy topological space. We proceed as
follows.
In Section 2, we present some basic definitions and ideas, also, we supply three lemmas
on the α-cuts of fuzzy subsets on a universe set X for all α in the interval [0,1]. We need
those lemmas for proofs scattered in subsequent sections.
In Section 3, we introduce our definition of fuzzy TL-uniform spaces, fuzzy uniform
map, and the fuzzy topology associated with a fuzzy TL-uniform space.
In Section 4, we study the relationship between a fuzzy TL-uniformity and its α-level
uniformities. Also, we generate fuzzy TL-uniformities from classical uniformities.
In Section 5, we prove that each category of fuzzy TL-uniform spaces and uniform
maps between them is a topological category, and we describe the fuzzy TL-uniform space
of the optimal lift of a source in this category. Also, we define a functor from the category
of fuzzy TL-uniform spaces into the category of fuzzy T-locality spaces, and we show that
it preserves optimal lifts.
Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2006, Article ID 25094, Pages 1–24
DOI 10.1155/IJMMS/2006/25094
2
Fuzzy TL-uniform spaces
2. Prerequisites
A triangular norm (cf. [1, 2]) is a binary operation on the unit interval I = [0,1] that is
associative, symmetric, isotone in each argument and has neutral element 1. A continuous triangular norm T is uniformly continuous, because its domain I × I is compact. This
means that for all ε > 0, there is θ = θT,ε > 0 in such a way that for every (α,β) ∈ I × I, we
have
(αTβ) − ε ≤ (α − θ)T(β − θ) ≤ αTβ ≤ (α + θ)T(β + θ) ≤ (αTβ) + ε.
(2.1)
Given a lower semicontinuous triangular norm T, the following binary operation on I:
jT (α,γ) = sup{θ ∈ I : αTθ ≤ γ},
(α,γ) ∈ I × I,
(2.2)
is called the residual implication of T [3], we often simplify jT to j. The best wellknown triangular norms are the simplest three, namely Min (also denoted by ∧), Tm (the
Lukasiewicz conjunction), and π (product), where for all α,β ∈ I,
αTm β = (α + β)∧1,
απβ = αβ,
(2.3)
where the binary operation ∧, above, is the truncated subtraction, defined on nonnegative real numbers by r ∧ s = max{r − s,0}, r,s ≥ 0.
Proposition 2.1 [4]. The T-residual implication j, of a lower semicontinuous triangular
norm T, enjoys the following properties for all α,β,γ ∈ I:
(IT1) αTβ ≤ γ if and only if α ≤ j(β,γ),
(IT2) αTβ > γ if and only if α > j(β,γ),
(IT3) j(α,γ) = 1 if and only if α ≤ γ,
(IT4) j(1,γ) = γ,
(IT5) j(αTβ,γ) = j(α, j(β,γ)),
(IT6) αT j(α,γ) ≤ γ,
(IT7) j(α,β)T j(θ,γ) ≤ j(αTθ,βTγ),
(IT8) j is antimonotone in the left argument and monotone in the right argument.
A fuzzy set λ in a universe X, introduced by Zadeh in [5], is a function λ : X → I. The
collection of all fuzzy subsets of X is denoted by I X . Ordinary subsets of a universe X will
frequently occur in what follows. We will often need to consider a subset M ⊆ X as a fuzzy
subset of X, said to be a crisp subset of X, which we will denote by the symbol 1M . We do
this by identifying 1M with its characteristic function. The collection of all crisp subsets
of X is denoted by 2X . The symbol α denotes the constant fuzzy set in a universe X with
value α ∈ I.
For the fuzzy sets μ,λ ∈ I X , the degree of containment of μ in λ according to j = jT is
the real number in I [6]:
j μ,λ = inf j μ(x),λ(x) .
x ∈X
(2.4)
K. A. Hashem and N. N. Morsi 3
Last, we denote by j |μ,λ| the following fuzzy subset of X:
j |μ,λ|(x) = j μ(x),λ(x) ,
x ∈ X.
(2.5)
Given a fuzzy set λ ∈ I X and a real number α ∈ I1 = [0,1[, the strong α-cut of λ is the
following subset of X:
λα = x ∈ X : λ(x) > α ,
(2.6)
and for a real number α ∈ I, the weak α-cut of λ is the subset of X:
λα∗ = x ∈ X : λ(x) ≥ α ,
(2.7)
for both strong α-cuts and weak α-cuts of λ, we identify crisp sets with their characteristic
functions.
It is direct to verify that every λ ∈ I X has the following resolutions:
λ=
α ∧ λα =
α∈I1
α ∧ λ α∗ .
(2.8)
α∈I0
We follow Lowen’s definition of a fuzzy interior operator on a set X [7].
This is an operator o : I X → I X that satisfies μo ≤ μ, (μ ∧ λ)o = μo ∧ λo for all μ,λ ∈ I X ,
and αo = α for all α ∈ I.
The pair (X,o ) is called a fuzzy topological space (FTS).
The category of all fuzzy topological spaces and continuous functions between them
(cf. [7]) is denoted by FTS.
Given a fuzzy topological space (X,o ) = (X,τ), and α ∈ I1 , the set
ια (τ) = λα ⊆ X : λ ∈ τ
(2.9)
is a topology on X, said to be the α-level topology of τ [8]. It is direct to see that its
interior operator is given by
intια (τ) (M) =
α ∨ 1M
o α
,
M ⊆ X,
(2.10)
and, ι(τ) = supα∈I1 ια (τ) is called the topology modification of τ [8].
Prefilters and prefilterbases were introduced by Lowen in [9]. A prefilter in a universe
/ , is closed under finite meets
X is a nonempty collection ⊂ I X which satisfies 0 ∈
and contains all the fuzzy supersets of its individual members. A prefilterbase in X is a
/ Ꮾ and the meets of two members of Ꮾ
nonempty collection Ꮾ ⊂ I X which satisfies 0 ∈
contain a member of Ꮾ. A prefilterbase Ꮾ is said to be prefilterbasis for a prefilter if
= [Ꮾ], where [Ꮾ] = {λ ∈ I X : there exists ν ∈ Ꮾ with ν ≤ λ}.
4
Fuzzy TL-uniform spaces
Definition 2.2 [10]. The T-saturation operator is the operator ∼ T which sends a prefilterbase Ꮾ in X to the following subset of I X :
∼T
Ꮾ
⎫⎤
⎡⎧
⎨ ⎬
⎣
=
γTμγ : μγ ∈ Ꮾ ∀γ ∈ [0,1[⎭⎦ ,
⎩
(2.11)
γ∈[0,1[
said to be the T-saturation of Ꮾ.
Theorem 2.3 [10]. The T-saturation operator ∼ T is isotone. Also, given a prefilterbase Ꮾ
in X, Ꮾ∼T satisfies
Ꮾ∼T is a prefilter, ⊇ [Ꮾ] ⊇ Ꮾ,
sup j υ,λ = sup j μ,λ,
υ ∈Ꮾ
μ ∈Ꮾ ∼ T
(2.12)
moreover, Ꮾ∼T equals the following subset of I X :
λ ∈ I X : j β,λ ∈ [Ꮾ] ∀β < 1 .
(2.13)
Definition 2.4 [10]. A fuzzy T-locality space is a fuzzy topological space (X,o ) whose fuzzy
interior operator is induced by some indexed family Ꮾ = (Ꮾ(x))x∈X , of prefilterbases in
I X , in the following manner:
μo (x) = sup j υ,μ,
υ∈Ꮾ(x)
∀(μ,x) ∈ I X × X.
(2.14)
The family Ꮾ is said to be a T-locality basis for (X,o ), the fuzzy topology of (X,o ) will be
denoted by τ(Ꮾ). The full subcategory, of FTS, of fuzzy T-locality spaces is denoted by
T-FLS.
Theorem 2.5 [10]. A family of prefilterbases in X, Ꮾ = (Ꮾ(x))x∈X , will be a T-locality base
in X if and only if it satisfies the following two conditions for all x ∈ X.
(TLB1) υ(x) = 1 for all υ ∈ Ꮾ(x).
(TLB2) Every υ ∈ Ꮾ(x) has a T-kernel. This consists of two families (υγ ∈ Ꮾ(x))γ∈I1 and
(υ yγθ ∈ Ꮾ(y))(y,γ,θ)∈X ×I1 ×I0 such that for all
(y,γ,θ) ∈ X × I1 × I0 ,
γTυγ (y) ∧θ Tυ yγθ ≤ υ,
where I0 =]0,1].
(2.15)
Theorem 2.6 [10]. (i) A family Ꮾ = (Ꮾ(x))x∈X will be a T-locality basis for a T-locality
space (X,o ) if and only if Ꮾ∼T is so. In this case, Ꮾ∼T is the greatest T-locality basis for (X,o )
(and Ꮾ∼T is called the T-locality system of (X,o )). It is the unique T-saturated T-locality
basis for (X,o ). In this case, Ꮾ is a basis for the T-locality system Ꮾ∼T .
(ii) If Ꮾ = (Ꮾ(x))x∈X is a T-locality basis for a T-locality space (X,o ), then Ꮾ∼T =
((Ꮾ(x))∼T )x∈X is obtained also by
Ꮾ(x)
∼T
= υ ∈ I X : υo (x) = 1 ,
x ∈ X.
(2.16)
A distance distribution function (DDF) [2] is a function from the set R+ of positive
real numbers to the unit interval I = [0,1], which is monotone, left continuous, and has
K. A. Hashem and N. N. Morsi 5
supremum 1. The set of all DDFs is denoted by D+ . Every continuous triangular norm
T induces on D+ a binary operation ⊕T by (η ⊕T ζ)(s) = sup{η(b)Tζ(s − b) : 0 < b < s},
s > 0. The identity element of a semigroup (D+ , ⊕T ) is ε0 defined by
⎧
⎨0,
ε0 (x) = ⎩
1,
x ≤ 0,
x > 0.
(2.17)
A probabilistic pseudometric on a set X (cf. [11]) is a function Γ : X × X → D+ that
satisfies for all x, y, z in X the following properties:
(PM1) Γ(x,x) = ε0 ,
(PM2) Γ(x, y) = Γ(y,x),
(PM3) Γ(x, y) ⊕T Γ(y,z) ≥ Γ(x,z).
In [12], Höhle defines for every ψ,ϕ ∈ I X ×X and λ ∈ I X , the T-section of ψ over λ
by ψ λT (x) = sup y∈X λ(y)Tψ(y,x), x ∈ X, the T-composition of ψ, ϕ by ψoT ϕ(x, y) =
supz∈X ϕ(x,z)Tψ(z, y), x, y ∈ X, the symmetric of ψ by s ψ(x, y) = ψ(y,x), x, y ∈ X.
Now, we will use the following lemmas in the sequel.
Lemma 2.7. For all ϕ, ψ ∈ I X ×X and α ∈ I1 ,
(i) (γTϕ)α = ϕ j(γ,α) for all γ ∈ I,
(ii) (ϕoT ψ)α =
θTβ>α (ϕθ ∗ oψβ∗ ) =
θ
β
θTβ≤α (ϕ oψ ).
Proof. (i) For every (x, y) ∈ X × X, we get the equivalences
α
(x, y) ∈ γTϕ
iff γTϕ(x, y) > α
iff ϕ(x, y) > j(γ,α),
by (IT2),
(2.18)
iff (x, y) ∈ ϕ j(γ,α) .
This proves (i).
(ii) For the first equality, let (x, y) ∈ X × X, we get the equivalences
(x, y) ∈ ϕoT ψ
iff
iff
α
ϕoT ψ (x, y) > α
sup ψ(x,z)Tϕ(z, y) > α
z ∈X
iff ∃z ∈ X,
β,θ ∈ Io with βTθ > α such that
(2.19)
ψ(x,z) ≥ β, ϕ(z, y) ≥ θ
iff ∃z ∈ X,
β,θ ∈ Io with βTθ > α such that
(x,z) ∈ ψβ∗ , (z, y) ∈ ϕθ∗
iff
(x, y) ∈
θTβ>α
which proves our assertion.
ϕθ∗ oψβ∗ .
6
Fuzzy TL-uniform spaces
For the second equality of (ii), we get also
(x, y) ∈
/ ϕoT ψ
iff
iff
α
ϕoT ψ (x, y) ≤ α
sup ψ(x,z)Tϕ(z, y) ≤ α
z ∈X
iff ∀z ∈ X, ∃β,θ ∈ I0 with βTθ ≤ α such that
(2.20)
ψ(x,z) ≤ β, ϕ(z, y) ≤ θ
iff ∀z ∈ X, ∃β,θ ∈ I0 with βTθ ≤ α such that
/ ϕθ
(x,z) ∈
/ ψ β , (z, y) ∈
iff
ϕθ oψ β .
(x, y) ∈
/
θTβ≤α
This completes the proof.
Lemma 2.8 [13]. For all ψ ∈ I X ×X and α ∈ I1 ,
sψ
α
= s ψα .
(2.21)
Lemma 2.9 [14]. Let (ψn )n∈N be a sequence of fuzzy sets on X × X for which ψn (x,x) = 1 for
all x ∈ X; ψn = s (ψn ); ψn+1 oT ψn+1 oT ψn+1 ≤ ψn . Then, there is a probabilistic pseudometric
Γ on X satisfying
ψn+1 (x, y) ≤ Γ(x, y) 2−n ≤ ψn (x, y),
∀(x, y) ∈ X × X.
(2.22)
3. Fuzzy TL-uniform spaces
The fuzzy TL-uniform spaces are introduced in this section, and some of their properties
are given. The uniform maps are defined. Also, we generate a fuzzy topology from a fuzzy
TL-uniform space.
Definition 3.1. (i) A fuzzy TL-uniform base on a set X is a subset υ ⊂ I X ×X which fulfills
the following properties.
(FLUB1) υ is a prefilterbase.
(FLUB2) For all ϕ ∈ υ and x ∈ X, ϕ(x,x) = 1.
(FLUB3) For all ϕ ∈ υ and γ ∈ I1 , there is ϕγ ∈ υ with γTϕγ ≤ s ϕ.
(FLUB4) For all ϕ ∈ υ and γ ∈ I1 , there is ϕγ ∈ υ with γT(ϕγ oT ϕγ ) ≤ ϕ.
Obviously, (FLUB3) and (FLUB4) can be replaced by the single condition
(FLUB3\ ). For all ϕ ∈ υ and γ ∈ I1 , there is ϕγ ∈ υ with γT(ϕγ oT ϕγ ) ≤ s ϕ.
(ii) A fuzzy TL-uniformity on X is a T-saturated fuzzy TL-uniform base on X.
(iii) If μ is a fuzzy TL-uniformity on X, then υ is a basis for μ if υ is a prefilterbase and
υ∼T = μ.
K. A. Hashem and N. N. Morsi 7
It follows that for a fuzzy TL-uniformity μ on a set X, and all ϕ ∈ μ, s ϕ ∈ μ. The pair
(X,μ) consisting of a set X and a fuzzy TL-uniformity μ on X is called fuzzy TL-uniform
space. The elements of μ are called fuzzy vicinities.
Proposition 3.2. Let υ be a fuzzy TL-uniform base on a set X, then the prefilter υ∼T
determined by υ is a fuzzy TL-uniformity on X.
Conversely, if μ is a fuzzy TL-uniformity on X, then conditions (FLUB2)–(FLUB4) are
satisfied by every prefilterbase υ which determined μ.
Proof. Suppose υ is a fuzzy TL-uniform base on X. We show that υ∼T is a fuzzy TLuniformity.
Obviously, υ∼T is T-saturated prefilter.
Now, to prove (FLUB2), let ϕ ∈ υ∼T , then there exists (ψα ∈ υ)α∈I1 such that ϕ ≥
α∈I1 (αTψα ). Hence for every x ∈ X, we have
ϕ(x,x) ≥
αTψα (x,x)
α∈I1
=
(αT1) for ψα ∈ υ
α∈I1
=
(3.1)
(α) = 1.
α∈I1
Also, to prove (FLUB3\ ), let ϕ ∈ υ∼T and γ ∈ I1 , then for all α ∈ I1 , there exists ψα ∈ υ
such that
ϕ ≥ αTψα .
(3.2)
But for each β ∈ I1 , there is β ψα ∈ υ with
β
βT ψα oT β ψα ≤ s ψα .
(3.3)
By continuity of T, we can choose β ∈ I1 such that βTβ = γ, and by taking ϕγ = β ψβ ∈
υ ⊆ υ∼T , we get
≤ βT s ψβ ,
by (3.3),
γT ϕγ oT ϕγ = βTβT β ψβ oT β ψβ
≤ s ϕ,
(3.4)
directly from (3.2).
This proves that υ∼T is a fuzzy TL-uniformity.
Conversely, suppose that μ is a fuzzy TL-uniformity, and let υ be a prefilterbase which
determines μ.
Since υ ⊆ μ, then obviously υ satisfies (FLUB2), while (FLUB3) follows immediately
from the fact that for all ψ ∈ μ, we get s ψ ∈ μ.
8
Fuzzy TL-uniform spaces
To prove (FLUB4), let ϕ ∈ υ and γ ∈ I1 , then ϕ ∈ μ and hence, there is a family (ψγ ∈
μ)γ∈I1 such that
γT ψγ oT ψγ ≤ ϕ.
(3.5)
By continuity of T, we can find α ∈ I1 such that γ ≤ αTαTα, and by Theorem 2.6, there
is ϕγ in υ with ϕγ ≤ j |α,ψα |, consequently,
γT ϕγ oT ϕγ ≤ αTαTαT j α,ψα oT j α,ψα = αT
≤ αT ψα oT ψα ,
≤ ϕ,
αT j α,ψα oT αT j α,ψα ,
clear,
(3.6)
by (IT6),
by (3.5),
which proves that υ satisfies (FLUB4), and completes the proof.
} ⊆ I X ×X
Lemma 3.3. If {ψα : α ∈ I1
and ψ =
X
ψ 1M T = α∈I1 [αTψα 1M T ] ∈ I .
α∈I1 (αTψα
) ∈ I X ×X ,
then for every M
∈ 2X ,
Proof. For every x ∈ X,
ψ 1M
T (x) = sup 1M (y)Tψ(y,x) = sup ψ(y,x)
y ∈X
y ∈M
⎫
⎧
⎨ ⎬
= sup
αTψα (y,x) ⎭
⎩
y ∈M
=
α∈I1
!"
αT sup ψα (y,x)
αTψα 1M
=
T
α∈I1
Hence ψ 1M T =
by isotonicity of T,
,
y ∈M
α∈I1
(3.7)
(x).
α∈I1 [αTψα 1M T ].
Lemma 3.4. For a prefilterbase
ψ 1M
T
#
in X × X and M ∈ 2X ,
:ψ ∈
∼T
$
"
= ψ 1M
T
:ψ ∈
#
$
"∼T
.
Proof. First, we show that {ψ 1M T : ψ ∈ ∼T } is T-saturated prefilter. Let ψ,ϕ ∈
and let μ ∈ I X be such that μ ≥ ψ 1M T ∧ ϕ1M T . Then we define φ ∈ I X ×X by
⎧
⎨μ(x)
φ(x, y) = ⎩
1
if x ∈ M,
if x ∈
/ M.
(3.8)
# ∼T
(3.9)
K. A. Hashem and N. N. Morsi 9
#
It is easy to see that μ = φ1M T and φ ≥ ψ ∧ ϕ and hence φ ∈ ∼T . This proves that
#
#
μ ∈ {ψ 1M T : ψ ∈ ∼T }, demonstrates that {ψ 1M T : ψ ∈ ∼T } is a prefilter.
# ∼T
# ∼T
Now, given a subfamily {ψγ : γ ∈ I1 } of
. Since
is T-saturated, α∈I1 (αTψα ) ∈
#∼T
, so,
αTψα 1M
α∈I1
T
=
αTψα
1M
α∈I1
∈ ψ 1M
:ψ ∈
T
T,
∼T
$
by Lemma 3.3,
"
(3.10)
.
This proves our assertion. Finally, we have
ψ 1M
T
:ψ ∈
$
⊆ ψ 1M
T
:ψ ∈
∼T
$
"
$
with γTψγ ≤ ψ
= ψ 1M T : ∀γ ∈ I1 ∃ψγ ∈
$
with γTψγ 1M T ≤ ψ 1M T
⊆ ψ 1M T : ∀γ ∈ I1 ∃ψγ ∈
$ ∼T
,
= ψ 1M T : ψ ∈
(3.11)
since the prefilter {ψ 1M T : ψ ∈
ψ 1M
T
# ∼T
:ψ ∈
} is T-saturated, then
$ ∼T
⊆ ψ 1M
⊆ ψ 1M
T
T
:ψ ∈
:ψ ∈
∼T
$
"
$ ∼T
(3.12)
.
So, equality holds, and winds up the proof.
Theorem 3.5. If υ is a fuzzy TL-uniform base on a set X, then the indexed family Ꮾ =
(Ꮾ(x))x∈X , given by
Ꮾ(x) = ϕ 1x
T
:ϕ∈υ ,
is a T-locality base on X.
(3.13)
Proof. Given x ∈ X and υ ∈ Ꮾ(x), then there is ϕ ∈ υ such that υ = ϕ1x T , hence
υ(x) = ϕ 1x
T (x) = sup
y ∈X
It follows that Ꮾ satisfies (TLB1).
1x (y) Tϕ(y,x) = 1x (x)Tϕ(x,x) = 1.
(3.14)
10
Fuzzy TL-uniform spaces
Also, for every υ ∈ Ꮾ(x), γ ∈ I1 , and z ∈ X, we have
υ(z) = ϕ 1x
T (z)
for some ϕ ∈ υ
= ϕ(x,z) ≥ γT ϕγ oT ϕγ (x,z),
= sup γTϕγ (x, y)Tϕγ (y,z)
by (FLUB4)
y ∈X
≥ sup γTϕγ (x, y)T δT ϕγδ oT ϕγδ (y,z) ,
y ∈X
= sup γTϕγ (x, y)TδT sup ϕγδ (y,s)Tϕγδ (s,z)
y ∈X
by (FLUB4) again
(3.15)
s∈X
≥ γTϕγ (x, y)TδTϕγδ (y,z)Tϕγδ (z,z)
= γTϕγ 1x T (y)Tδ T ϕγδ 1 y T (z) .
Now, for every θ ∈ I0 , we can get (by continuity of T) δ = δθ ∈ I1 for which [γTϕγ 1x T
(y)Tδ] ≥ [(γTϕγ 1x T (y))∧θ], hence take υxγ = ϕγ 1x T ∈ Ꮾ(x) and υ yγθ = ϕγδ 1 y T ∈
Ꮾ(y) which satisfy υ(z) ≥ [(γTυxγ (y))∧θ]Tυ yγθ (z).
This shows that Ꮾ satisfies (TLB2), which completes the proof.
Proposition 3.6. If μ is a fuzzy TL-uniformity on a set X, then the indexed family Ꮾ =
(Ꮾ(x))x∈X , given by
Ꮾ(x) = ϕ 1x
T
:ϕ∈μ ,
is a T-locality system on X.
(3.16)
The proof follows at once from Theorem 3.5 and Lemma 3.4.
We will see how a fuzzy TL-uniform base can generate a fuzzy topology.
Definition 3.7. The fuzzy T-locality space (X,τ T (υ)) induced by a fuzzy TL-uniform base
υ on X is the T-FLS defined by means of the fuzzy T-locality base of the preceding theorem, for which its fuzzy interior operator o is given by
% λo (x) = sup j ϕ 1x
ϕ ∈υ
T ,λ
&
,
(λ,x) ∈ I X × X.
(3.17)
The fuzzy topology τ T (υ) of that space is also denoted simply by τ(υ). Notice that due
to Theorem 1.4, a fuzzy TL-uniform base υ and its fuzzy TL-uniformity υ∼T induce the
same fuzzy T-locality space, that is, τ(υ) = τ(υ∼T ).
Uniform-type continuity:
Definition 3.8. Let (X,μ) and (X \ ,μ\ ) be fuzzy TL-uniform spaces and f : X → X \ . f is
a fuzzy uniform map (or fuzzy uniformly continuous) if any of the following equivalent
conditions hold.
(i) For each ψ ∈ μ\ , ( f × f )−1 (ψ ) ∈ μ.
(ii) For each ψ ∈ μ\ , there is ψ ∈ μ such that ψ ≤ ( f × f )−1 (ψ ).
(iii) For each ψ ∈ μ\ , there is ψ ∈ μ such that ( f × f )(ψ) ≤ ψ ,
where ( f × f )−1 (ψ )(x, y) = ψ ( f (x), f (y)), x, y ∈ X.
K. A. Hashem and N. N. Morsi 11
The composite of two uniformly continuous functions f : (X,μ) → (Y ,U) and g :
(Y ,U) → (Z, ℘) is again uniformly continuous, since (g f × g f )−1 (ψ) = ( f × f )−1 {(g ×
g)−1 (ψ)} ∈ μ, for all ψ ∈ ℘.
Proposition 3.9. If (X,μ) and (X \ ,μ\ ) are fuzzy TL-uniform spaces, with bases υ and υ\ ,
respectively, and f : X → X \ , then f is uniformly continuous if and only if for all ϕ ∈ υ\ and
all γ ∈ I1 , there is ϕ ∈ υ such that
γTϕ ≤ ( f × f )−1 (ϕ ).
(3.18)
Proof. Let f : (X,μ) → (X \ ,μ\ ) be uniformly continuous and ϕ ∈ υ\ , γ ∈ I1 . Then ϕ ∈
υ\∼T = μ\ , since f is uniformly continuous, we get ( f × f )−1 (ϕ ) ∈ μ = υ∼T , hence there
is ϕ ∈ υ such that γTϕ ≤ ( f × f )−1 (ϕ ).
Conversely, suppose that the stated condition holds.
Let ψ ∈ μ\ , since μ\ = υ\∼T , then for all γ ∈ I1 , there exists ϕγ ∈ υ\ such that ψ ≥
γTϕγ , hence
( f × f )−1 (ψ ) ≥ ( f × f )−1 γTϕγ = γT ( f × f )−1 ϕγ
≥ γTθTϕγθ
for some θ ∈ I1 , ϕγθ ∈ υ, by hypothesis.
(3.19)
Since T is continuous, then for all α ∈ I1 , there exists γ = γα ∈ I1 such that α = γTγ, by
taking φα = ϕγγ ∈ υ, we get
( f × f )−1 (ψ) ≥ γTγTϕγγ = αTφα ,
(3.20)
which implies that ( f × f )−1 (ψ) ∈ μ. Hence f is uniformly continuous, and winds up
the proof.
A function f : (X, o ) = (X,τ) → (Y , o ) = (Y ,τ \ ), between two fuzzy topological spaces,
is said to be continuous [7]; if f −1 (μ) ∈ τ for all μ ∈ τ \ , μ ∈ I Y . Equivalently, if
o
f −1 (λo ) ≤ [ f −1 (λ)] for all λ ∈ I Y .
Theorem 3.10. Let (X,μ) and (Y , ℘) be fuzzy TL-uniform spaces. If f : X → Y is uniformly
continuous, then it is continuous with respect to the fuzzy topologies generated by μ and ℘,
respectively.
Proof. Let λ ∈ I Y and x ∈ X, we denote by o1 and o2 the fuzzy interior operators of τ(μ)
and τ(℘), respectively. Then, we have
% f −1 λo2 (x) = λo2 f (x) = sup j ψ 1 f (x)
ψ ∈℘
= sup inf j ψ 1 f (x)
ψ ∈℘ y ∈Y
T ,λ
&
T (y),λ(y)
≤ sup inf j ψ f (x), f (z) ,λ f (z)
ψ ∈℘ z∈X
for range f ⊆ Y
12
Fuzzy TL-uniform spaces
= sup inf j ( f × f )−1 (ψ)(x,z), f −1 (λ)(z)
ψ ∈℘ z∈X
≤ sup inf j ϕ(x,z), f −1 (λ)(z) ,
ϕ ∈μ z ∈X
by uniformly continuous of f and (IT8),
= sup inf j ϕ 1x T (z), f −1 (λ) (z)
ϕ ∈μ z ∈X
o1
= f −1 (λ) (x),
(3.21)
which proves the continuity of f : (X,τ(μ)) → (Y ,τ(℘)) at each point x in X.
4. The operators ιu,α and ωu
Now, we introduce the concept of α-level uniformities for a given fuzzy TL-uniformity
and we study the relationships between them.
For a fuzzy TL-uniformity μ on a set X and α ∈ I1 , we define
ιu,α (μ) = ψ j(β,α) ⊆ X × X : ψ ∈ μ, β ∈]α,1] ,
(4.1)
we next show that ιu,α (μ) is a uniformity on X whenever μ is a fuzzy TL-uniformity, called
the α-level uniformity of μ.
In the following two propositions, let υ be a basis for a fuzzy TL-uniformity μ on a
nonempty set X.
Proposition 4.1. ιu,α (υ) is a filterbasis for the filter ιu,α (μ).
Proof. We first show that ιu,α (υ) is indeed a filterbase.
/ ιu,α (υ), because every member in ιu,α (υ) contains the diagonal of X(D(X)).
(a) ∅ ∈
(b) The intersection of two members of ιu,α (υ) contains a member: given U,V ∈
ιu,α (υ), then there are ψ,ϕ ∈ υ and β1 ,β2 ∈]α,1] such that U = ϕ j(β1 ,α) , V = ψ j(β2 ,α) , without loss of generality, β1 ≤ β2 . So
U ∩ V = ϕ j(β1 ,α) ∩ ψ j(β2 ,α)
⊇ ϕ j(β1 ,α) ∩ ψ j(β1 ,α) ,
= (ϕ ∧ ψ)
j(β1 ,α)
because j β2 ,α ≤ j β1 ,α
(4.2)
.
But, there is ζ ∈ υ with ζ ≤ ϕ ∧ ψ (because υ is a prefilterbase). Thus U ∩ V contains a
member ζ j(β1 ,α) of ιu,α (υ).
To prove that ιu,α (μ) is a filter, it now suffices to show that ιu,α (μ) is closed under supersets: given U ∈ ιu,α (μ), let V ⊇ U. Then
U = ψ j(β,α)
for some ψ ∈ μ, β ∈]α,1].
(4.3)
K. A. Hashem and N. N. Morsi 13
If we take ϕ = V
and
j(β,α) ∈ I X ×X , we get ϕ ≥ ψ j(β,α)
V = V ∨ j(β,α)
j(β,α)
j(β,α) ≥ ψ therefore, ϕ is also in μ
= ϕ j(β,α) ∈ ιu,α (μ),
(4.4)
which proves our assertion.
Finally, we show that ιu,α (μ) is generated by ιu,α (υ): let U ∈ ιu,α (μ), then U = ψ j(β,α) for
some ψ ∈ μ and β ∈]α,1]. Since υ ⊆ υ∼T ⊂ μ, then for all γ ∈ I1 , there exists ϕγ ∈ υ with
γTϕγ ≤ ψ.
(4.5)
By continuity of T, we can choose γo ∈ I1 such that γo Tβ ∈]α,1]. Then
ϕγo
j(γo Tβ,α)
j(γo , j(β,α))
= ϕγo
,
j(β,α)
= γo Tϕγo
,
⊆ ψ j(β,α) ,
by (IT5),
by Lemma 2.7(i),
(4.6)
by (4.5),
= U.
Thus there is a member (ϕγo ) j(γo Tβ,α) of ιu,α (υ) contained in U. Since also ιu,α (υ) is included in ιu,α (μ).
This completes the proof that it is a filterbasis for ιu,α (μ).
Proposition 4.2. ιu,α (υ) is a uniform basis for the uniformity ιu,α (μ).
Proof. ιu,α (υ) is a uniform base.
(UB1) ιu,α (υ) is a filterbase (from Proposition 4.1).
(UB2) Each member of ιu,α (υ) evidently contains the diagonal D(X): since ϕ(x,x) =
1 > j(β,α) for all (ϕ,x,β) ∈ υ × X ×]α,1], then (x,x) ∈ ϕ j(β,α) for all (ϕ j(β,α) ,x) ∈ (ιu,α (υ))
× X.
(UB3) For every U ∈ ιu,α (υ), its symmetric s U contains a member of ιu,α (υ): given
U ∈ ιu,α (υ), then there exist ϕ ∈ υ and β ∈]α,1], such that U = ϕ j(β,α) , hence for all γ ∈ I1 ,
there exists ϕγ ∈ υ with
γTϕγ ≤ s ϕ.
(4.7)
By continuity of T, we can get γo ∈ I1 such that γo Tβ ∈]α,1]. Then
ϕγo
j(γo Tβ,α)
j(γo , j(β,α))
= ϕγo
,
j(β,α)
= γo Tϕγo
,
j(β,α)
⊆ (s ϕ)
,
j(β,α) =s ϕ
,
by (IT5),
by Lemma 2.7(i),
by (4.7),
by Lemma 2.8,
= s U.
Therefore, s U contains the member (ϕγo ) j(γo Tβ,α) of ιu,α (υ).
(4.8)
14
Fuzzy TL-uniform spaces
(UB4) For every U ∈ ιu,α (υ), there exists V ∈ ιu,α (υ) such that V oV ⊂ U: given U ∈
ιu,α (υ), there are ϕ ∈ υ and β ∈]α,1], such that U = ϕ j(β,α) . Also, for all γ ∈ I1 , there exists
ϕγ ∈ υ with
γT ϕγ oT ϕγ ≤ ϕ.
(4.9)
By continuity of T, we can choose γo ∈ I1 such that γo ≥ β and γo Tβ ∈]α,1]. Then
U = ϕ j(β,α) ⊇ γ T ϕγo oT ϕγo
j(β,α)
o
j(γo , j(β,α))
= ϕγo oT ϕγo
, by Lemma 2.7(i),
j(γo Tβ,α)
= ϕγo oT ϕγo
, by (IT5),
θ δ =
ϕγo o ϕγo , by Lemma 2.7(ii),
θTδ ≤ j(γo Tβ,α)
⊇
θ ϕγo o ϕγo
δ ,
(4.10)
by (IT8),
θTδ ≤ j(γo Tβ,αTα)
j(γo ,α) j(β,α)
⊇ ϕγo
o ϕγo
, by (IT7),
j(β,α) j(β,α)
⊇ ϕγo
o ϕγo
, by (IT8) for γo ≥ β.
Taking V = (ϕγo ) j(β,α) , which is in ιu,α (υ), proves our assertion. This completes the proof
that ιu,α (υ) is a uniform base.
Consequently, by Proposition 4.1, ιu,α (μ) is uniformity with ιu,α (υ) a basis.
Corollary 4.3. Let (X,μ) be a fuzzy TL-uniform space, in the cases T = Tm , min, the level
uniformities of (X,μ) form an ascending chain, while in the case T = π, the level uniformities
form an antichain.
Proof. Let 0 ≤ α1 < α2 < 1 and V ∈ ιu,α1 (μ).
Then there are ϕ ∈ μ and β ∈]α1 ,1] such that V = ϕ j(β,α1 ) . First, whenever T = Tm , we
have
j β,α1 = min 1 − β + α1 ,1 ,
by (2.2),
= min 1 − β + α1 − α2 + α2 ,1
= j α2 + β − α1 ,α2 ,
(4.11)
hence, V = ϕ j(β,α1 ) = ϕ j(α2 +β−α1 ,α2 ) ∈ ιu,α2 (μ), because it is easy to see that α2 + β − α1 ∈
]α2 ,1], since β > α1 , which proves that ιu,α1 (μ) ⊆ ιu,α2 (μ).
Second, whenever T = Min, we have
j(β,α) = α ∀β > α, by (2.2).
(4.12)
K. A. Hashem and N. N. Morsi 15
Hence
V = ϕ j(β,α1 ) = ϕα1
⊇ ϕα2 ,
=ϕ
because α1 < α2 ,
j(γ,α2 )
(4.13)
for any γ ∈ α2 ,1 .
But for every γ ∈]α2 ,1], ϕ j(γ,α2 ) ∈ ιu,α2 (μ), V ∈ ιu,α2 (μ), because ιu,α2 (μ) is a filter. Hence
ιu,α1 (μ) ⊆ ιu,α2 (μ).
Finally, whenever T = π, we have
j(β,α) =
α
β
∀β > α, by (2.2).
(4.14)
Let W ∈ ιu,α2 (μ). Then W = ψ j(βo ,α2 ) for some ψ ∈ μ and βo ∈]α2 ,1]. For this number βo ,
take γ0 = α1 (βo /α2 ), we get
W = ψ j(βo ,α2 ) = ψ (α2 /βo ) = ψ (α1 /γ0 ) = ψ j(γo ,α1 ) ∈ ιu,α1 (μ)
(4.15)
because, obviously, we can see that γo ∈]α1 ,1], which proves that ιu,α2 (μ) ⊆ ιu,α1 (μ).
Proposition 4.4. If (X,μ) and (Y , ℘) are fuzzy TL-uniform spaces, and f : X → Y is a
uniform map, then it is also a uniform map when considered as a function between the
uniform spaces (X,ιu,α (μ)) and (Y ,ιu,α (℘)) for all α ∈ I1 .
The proof immediately follows from the definition of α-level uniformities and the fact that
( f × f )−1 (ψ β ) = (( f × f )−1 (ψ))β for all ψ ∈ ℘, β ∈ I1 .
For every fixed α ∈ I1 , define a function ι∼
u,α from the category of fuzzy TL-uniform
spaces and fuzzy uniform maps to the category of uniform spaces and uniform maps by
on objects : ι∼
u,α
on
morphisms : ι∼
u,α
is ιu,α ,
is the identity function.
(4.16)
Then an obvious conclusion from the above proposition is that these ι∼
u,α are well-defined
functors.
Now, for a uniformity u on a set X, we define
ωu (u) = ψ ∈ I X ×X : ∀γ ∈ I1 , ψ γ ∈ u ,
(4.17)
we supply the proof that ωu (u) is a fuzzy TL-uniformity on X.
Proposition 4.5. If u is a uniformity on a set X, then ωu (u) is a fuzzy TL-uniformity with
u as a basis.
Proof. First, it is easy to see that ωu (u) is a fuzzy TL-uniform base. To show that ωu (u) is
T-saturated, let ψ ∈ (ωu (u))∼T . Then
∀γ ∈ I1
∃ϕγ ∈ ωu (u)
with ψ ≥ γTϕγ .
(4.18)
16
Fuzzy TL-uniform spaces
For every α ∈ I1 , choose γ = γα > α in I1 for which
ψ α ⊇ γTϕγ
α
j(γ,α)
= ϕγ
,
(4.19)
since (ϕγ ) j(γ,α) ∈ u, because j(γ,α) ∈ I1 , then ψ α ∈ u because u is a filter, hence ψ ∈
ωu (u). This shows that (ωu (u))∼T = ωu (u).
Second, to show that u is a basis for ωu (u), we prove that (u)∼T = ωu (u): let ψ ∈ (u)∼T .
Then there is a family (Wβ ∈ u)β∈I1 such that
ψ ≥ βT1Wβ .
(4.20)
Now, for every α ∈ I1 , we choose γ > α in I1 for which we get
ψ α ⊇ γT1Wγ
α
α
= γ ∧ 1W γ
for 1Wγ is crisp
(4.21)
= 1Wγ = Wγ ∈ u,
thus ψ α ∈ u, because u is filter, consequently, ψ ∈ ωu (u) renders (u)∼T ⊆ ωu (u).
On the other hand, if ϕ ∈ ωu (u), then
ϕ=
α ∧ ϕα ≥
α∈I1
αTϕα .
(4.22)
α∈I1
Since ϕα ∈ u, for all α ∈ I1 , we get ϕ ∈ (u)∼T , therefore, ωu (u) ⊆ (u)∼T , and hence equal
ity holds.
Proposition 4.6. If (X,u) and (Y ,ᐃ) are uniform spaces, and f : X → Y is uniformly
continuous in the usual sense, then it is uniformly continuous when considered as a function
between the fuzzy TL-uniform spaces (X,ωu (u)) and (Y ,ωu (ᐃ)).
Now, if we denote by TL-FUS (US) the category of fuzzy TL-uniform spaces (the
category of uniform spaces) together with the uniformly continuous functions between
these spaces, and define the map ω∼ : US → TL-FUS by setting ω∼ (X,u) = (X,ωu (u))
and ω∼ ( f ) = f , we get that ω∼ is a well-defined functor.
In the following, we denote the fuzzy topology associated with a fuzzy TL-uniformity
μ by τ(μ) and the topology associated with a uniformity u by T(u).
Now, we introduce some compatibility between the above notions.
Theorem 4.7. (i) The α-level topology ια (τ(μ)) of the fuzzy T-locality space (X,τ(μ)), coincides with the topology T(ιu,α (μ)) on X, induced by the α-level uniformity ιu,α (μ) of a fuzzy
TL-uniform space (X,μ).
(ii) For a uniform space (X,u), the fuzzy topology τ(ωu (u)) induced by the above fuzzy
TL-uniformity ωu (u) is the same fuzzy topology ω(T(u))
∼
(iii) ι∼
u,α oω = IdUS for all α ∈ I1 .
K. A. Hashem and N. N. Morsi 17
Proof. (i) Let o be the fuzzy interior operator of the T-FLS (X,τ(μ)). Then for every
nonempty M ∈ 2X ,
intια(τ(μ)) M =
α∨M
o α
,
by (2.10),
o
= x : α ∨ M (x) > α
(
'
= x : sup inf j ψ 1x T (y), α ∨ M (y) > α ,
ψ ∈μ y ∈X
by Definition 2.4,
= x : ∃ψ ∈ μ, β ∈]α,1[ s.t. ∀ y ∈
/ M, j ψ(x, y),α ≥ β
/ M, βTψ(x, y) ≤ α , by (IT1),
= x : ∃ψ ∈ μ, β ∈]α,1[ s.t. ∀ y ∈
α = x : ∃ψ ∈ μ, β ∈]α,1[ s.t. ∀ y ∈
/ M, (x, y) ∈
/ βTψ
(4.23)
/ M, (x, y) ∈
/ ψ j(β,α)
= x : ∃ψ ∈ μ, β ∈]α,1[ s.t. ∀ y ∈
/ M, (x, y) ∈
/V
= x : ∃V ∈ ιu,α (μ) s.t. ∀ y ∈
/ M, y ∈
/ V x = x : ∃V ∈ ιu,α (μ) s.t. ∀ y ∈
= x : V x ⊆ M for some V ∈ ιu,α (μ)
= intT(ιu,α (μ)) M.
This demonstrates that ια (τ(μ))) = T(ιu,α (μ)), which renders (i).
(ii) Let the fuzzy set λ be τ(ω(u))-open.
If x ∈
/ (intT(u) λε ) for some x ∈ X and ε ∈ I1 , we get V x ⊂ λε , for all V ∈ u, thus, there
/ λε , that is, there exists z ∈ X such
exists z ∈ X such that z ∈ V x for all V ∈ u with z ∈
that z ∈ V x for all V ∈ u with λ(z) ≤ ε, hence
λ(x) = intτ(ω(u)) λ (x),
by hypothesis
= sup inf j ψ 1x
ψ ∈ω(u) y ∈X
T (y),λ(y)
= sup inf j 1V 1x T (y),λ(y) ,
V ∈u y ∈X
≤ sup j 1V 1x T (z),λ(z)
by Proposition 4.4,
(4.24)
V ∈u
= j 1,λ(z)
= λ(z),
by (IT4),
≤ ε.
That is, x ∈
/ λε , which shows that λε ⊆ intT(u) λε , ε ∈ I1 . Consequently, λε = intT(u) λε , that
ε
is, λ ∈ T(u) for all ε ∈ I1 .
18
Fuzzy TL-uniform spaces
On the other hand, let λ be ω(T(u))-open.
Given x ∈ X, then for all ε ∈ I1 with ε < λ(x), we have x ∈ λε , and hence, by hypothesis
x ∈ intT(u) λε , thus there exists W ∈ u such that W x ⊂ λε , that is, there exists W ∈ u
such that for all z ∈ W x, we get z ∈ λε , that is, there exists W ∈ u such that for all
z ∈ W x, we get λ(z) > ε. Hence,
intτ(ω(u)) λ (x) = sup inf j ψ 1x
ψ ∈ω(u) y ∈X
T (y),λ(y)
= sup inf j 1V 1x T (y),λ(y)
(4.25)
V ∈u y ∈X
≥ inf j 1W 1x T (y),λ(y) ≥ ε.
y ∈X
This shows that λ ≤ intτ(ω(u)) λ, which proves that λ is τ(ωu (u))-open, and winds up the
proof of (ii).
(iii) Follows immediately from the definitions.
Example 4.8. Let (X,<) be an ordered set. For each point xo ∈ X, let Vxo = {(x, y) : x, y >
xo } and define Uxo = D(X) ∪ Vxo . Then the reader can easily check that the collection
u∗ = {Ux : x ∈ X } is a basis for a uniformity u on X. Hence by the above notions, we
have
ωu (u) = ψ ∈ I X ×X : ∀γ ∈ I1 , ψ γ ∈ u
(4.26)
is a fuzzy TL-uniformity on X with u as a basis. Moreover,
ιu,α ωu (u) = ψ j(β,α) ⊆ X × X : ω ∈ ωu (u), β ∈]α,1] = u.
(4.27)
Definition 4.9. A fuzzy topological space (X,τ) is called a fuzzy TL-uniformizable if there
is a fuzzy TL-uniformity μ on X such that τ = τ(μ).
Corollary 4.10. A topological space (X,T) is uniformizable if and only if (X,ω(T)) is
fuzzy TL-uniformizable.
Proof. If T = T(u), then
ω(T) = ω T(u) = τ ω(u) .
(4.28)
Conversely, let ω(T) = τ(μ), then
T = ι ω(T) = sup ια ω(T) ,
α∈I1
= sup ια τ(μ) ,
α∈I1
= sup T ιu,α (μ) ,
α∈I1
)
*
= T sup ιu,α (μ) .
α∈I1
by definition of modification topology,
by hypothesis,
by Theorem 4.7(i),
(4.29)
K. A. Hashem and N. N. Morsi 19
Definition 4.11. A T-fls (X,τ(Ꮾ)) is said to be the following.
(i) L-T0 , if for every x = y in X, there is υ ∈ Ꮾ(x) ∪ Ꮾ(y) such that
υ(x) ∧ υ(y) < 1.
(4.30)
(ii) L-regular, if for every (M,x,ε) ∈ 2X × X × I0 , there is υ ∈ Ꮾ(x) with sup y∈X (υ
∧ 1M )(y) < ε, then there are an open set μ and ρ ∈ Ꮾ(x) such that 1M ≤ μ and
sup y∈X (μTρ)(y) < ε.
Theorem 4.12. TL-uniformizability ⇒ L-regularity.
Proof. Suppose (X,μ) is a fuzzy TL-uniform space.
Let M ∈ 2X , x ∈ X, and ε ∈ I0 are such that there is υ ∈ Ꮾ(x) with
sup υ ∧ 1M (y) < ε,
(4.31)
y ∈X
consequently, we can find ε0 very small such that
sup υ ∧ 1M (y) + ε0 < ε.
y ∈X
(4.32)
Since, υ ∈ Ꮾ(x), then for all γ ∈ I1 there are ψ,ψγ ∈ μ such that υ = ψ 1x T and ψ ≥
γT(ψγ oT ψγ ), (by using (FLUB4)). Hence
ε > sup ψ 1x
z ∈X
∧
1
(z) + ε0
M
T
≥ sup γT ωγ oT ψγ 1x T (z) + ε0
z ∈M
= sup γT ψγ oT ψγ (x, y) + ε0
z ∈M
'
= γT sup sup ψγ (x, y)Tψγ (y,z)
(
z ∈M y ∈X
+ ε0
(4.33)
'
(
= γT sup sup ψγ (x, y)T s ψγ (z, y) + ε0
y ∈X z ∈M
⎡
≥ (γ + θ)T sup ⎣ψγ 1x
y ∈X
T (y)T
z ∈M
s ψγ
1z
T (y)
⎤
⎦,
θ = θT,ε0 > 0 as in (2.1).
20
Fuzzy TL-uniform spaces
Choosing γ0 ∈ I1 for which (γ0 + θ) = 1, and taking ρ = s ψγ0 1x T , μ = ( z∈M s ψγ0 1z T )o ,
we get ρ ∈ Ꮾ(x), and μ is open, with ε > sup y∈X [ρ(y)Tμ(y)], also μ ≥ 1M , because for every x ∈ M, we have
⎛
μ(x) = ⎝
≥
s ψγ0
1z
z ∈M
s ψγ0
o
o
1x
= υx (x)
= 1,
⎞o
T
T
⎠ (x)
(4.34)
(x)
for s ψγ0 ∈ μ
by Theorem 2.3.
This proves the L-regularity of (X,τ(μ)).
Theorem 4.13. If (X,μ) is a fuzzy TL-uniform space, then (X,τ(μ)) is L-T0 if and only if
(inf ψ ∈μ ψ)1∗ = D(X).
Proof. A T-FLS (X,τ(μ)) is L-T0
iff ∀x = y in X, ∃υ ∈ Ꮾ(x) ∪ Ꮾ(y) such that υ(x) ∧ υ(y) < 1
iff
iff
∀x = y in X, ∃ψ ∈ μ such that ψ 1x T (x) ∧ ψ 1x T (y) < 1
∀x = y in X, ∃ψ ∈ μ such that ψ(x, y) = ψ 1x T (y) < 1
iff ∀x = y in X, ∃ψ ∈ μ such that (x, y) ∈
/ ψ1∗
iff
inf
ψ ∈μ 1∗
=
/
ψ ∈μ
(4.35)
ψ1∗ = D(X).
Theorem 4.14. Let Γ be a probabilistic pseudometric on a set X. Then there is a fuzzy TLuniformity μ = μ(Γ) given by
μ(Γ) = ψ ∈ I X ×X : ∀γ ∈ I1 ∃n ∈ N with γTΓ(x, y) 2−n ≤ ψ(x, y) .
(4.36)
Proof. Obviously, μ(Γ) is T-saturated prefilter.
Now, if we put ψn (x, y) = Γ(x, y)(2−n ), we get a sequence (ψn )n∈N satisfies ψn (x,x) = 1,
ψn = s (ψn ), and ψn+1 oT ψn+1 ≤ ψn , which shows that μ(Γ) is a fuzzy TL-uniformity.
Remark 4.15. Let μ be a fuzzy TL-uniformity on X, a nonempty subset Ꮾ of μ is said to be
a base of μ if for each ψ ∈ μ and each n ∈ N, there is ϕn ∈ Ꮾ such that (1 − 1/n)Tϕn ≤ ψ.
Theorem 4.16. A fuzzy TL-uniformity μ on a set X is probabilistic pseudo-metrizable if
and only if it has a countable base.
Proof. Let μ be probabilistic pseudo-metrizable. Then by using (PM1) and Theorem 4.14,
we get the necessary condition. For the sufficiency of the condition, suppose that μ has a
countable base Ꮾ = {ψ1 ,ψ2 ,...,ψn ,...}.
We set ϕ1 = ψ1 ∧ s ψ1 , then ϕ1 is a symmetric member of μ.
K. A. Hashem and N. N. Morsi 21
For a fixed γ in I1 , apply (FLUB4) twice for the member ϕ1 ∧ ψ2 which is in μ, then
there is a symmetric member say ϕ2 in μ such that
γT ϕ2 oT ϕ2 oT ϕ2 ≤ ϕ1 ∧ ψ2 .
(4.37)
Next, consider ϕ2 ∧ ψ3 and get a symmetric ϕ3 in μ such that
γT ϕ3 oT ϕ3 oT ϕ3 ≤ ϕ2 ∧ ψ3 .
(4.38)
In this manner, we proceed by induction a sequence (ϕn )n∈N of fuzzy vicinities for which
ϕn = s ϕn ,
γT ϕn+1 oT ϕn+1 oT ϕn+1 ≤ ϕn ∧ ψn+1 ≤ ϕn ,
(4.39)
by using Lemma 2.9, there is a probabilistic pseudometric Γ on X, with
γTϕn+1 (x, y) ≤ Γ(x, y) 2−n ≤ ϕn (x, y);
that is, the fuzzy TL-uniformity μ is generated by Γ, which proves our assertion.
(4.40)
5. Optimal lifts in fuzzy TL-uniform spaces
We prove that the category of fuzzy TL-uniform spaces and uniform maps between them
is a topological category, by constructing optimal lifts of sources in it. For expansion on
the categorical notions mentioned below, see [15].
The category FTS of fuzzy topological spaces and their continuous functions is a topological category. This means that its objects are sets with structures (here, fuzzy topologies), its morphisms are admissible functions between those sets (here, the continuous
functions), morphism composition and the identity morphisms are the usual ones for
functions, and the following three conditions are satisfied for all nonempty sets X.
(1) Every source ( f j : X → (X j ,σ j ) j ∈J ) of functions in the category FTS has a unique
optimal lift (also called initial lift), namely, the coarsest structure on X making
each function f j a morphism. Specifically, the optimal lift of that source is the
fuzzy topology τ on X with subbasis [16]:
τ ∗ = f j−1 (λ) ∈ I X : j ∈ J, λ ∈ σ j ,
(5.1)
that is, τ is the smallest fuzzy topology on X that contains τ ∗ .
(2) The class FTS(X), of all FTS-structures (i.e., fuzzy topologies) on X, is a set
(smallness condition).
(3) If X is a singleton, then FTS(X) is a singleton.
An important construction is that of initial fuzzy TL-uniformities. Given a family ( f j :
X → (X j ,μ j )) j ∈J , where for each j ∈ J, (X j ,μ j ) is a fuzzy TL-uniform space and f j is a
function from some set X to X j , we want to construct on X (in accordance with the
general definition of initial lifts in categories of sets with structures) a coarsest fuzzy TLuniformity making each function f j uniformly continuous. This fuzzy TL-uniformity is
called the initial fuzzy TL-uniformity on X for the family ( f j : X → (X j ,μ j )) j ∈J .
22
Fuzzy TL-uniform spaces
If (Y , ℘) is a fuzzy TL-uniform space and h : Y → X, then if X carries the initial fuzzy
TL-uniformity, h is uniformly continuous if and only if each f j oh is uniformly continuous.
Lemma 5.1. For a fuzzy TL-uniform base υ on a set Y and a function f : X → Y ,
( f × f )−1 υ∼T ⊆ ( f × f )−1 (υ)
∼T
∼T
= ( f × f )−1 (ϕ) : ϕ ∈ υ
.
(5.2)
Proof. Let ψ ∈ ( f × f )−1 (υ∼T ) and γ ∈ I1 . Then there are ϕ ∈ υ∼T and ϕγ ∈ υ, with ψ =
( f × f )−1 (ϕ) and γTϕγ ≤ ϕ.
By taking ψγ = ( f × f )−1 (ϕγ ) ∈ ( f × f )−1 (υ), we have
γTψγ = γT( f × f )−1 ϕγ = ( f × f )−1 γTϕγ ≤ ( f × f )−1 (ϕ) = ψ.
(5.3)
Therefore, ψ ∈ [( f × f )−1 (υ)]∼T , which completes the proof of the lemma.
Proposition 5.2. Let f : X → Y be a function. If υ is a fuzzy TL-uniform base on Y , then
( f × f )−1 (υ) is a fuzzy TL-uniform base on X.
Proof. (FLUB1) ( f × f )−1 (υ) is indeed a prefilterbase.
(FLUB2) For all ψ ∈ ( f × f )−1 (υ) and x ∈ X, we have
ψ(x,x) = ( f × f )−1 (ϕ) (x,x)
= ϕ f (x), f (x)
for some ϕ ∈ υ
(5.4)
= 1.
(FLUB3\ ) Let ψ ∈ ( f × f )−1 (υ) and γ ∈ I1 , we get ϕ and ϕγ in υ such that
ψ = ( f × f )−1 (ϕ),
γT ϕγ oT ϕγ ≤ s ϕ.
(5.5)
By taking ψγ = ( f × f )−1 (ϕγ ) ∈ ( f × f )−1 (υ), we have
γT ψγ oT ψγ = γT ( f × f )−1 ϕγ oT ( f × f )−1 ϕγ
= γT ( f × f )−1 ϕγ oT ϕγ
≤ ( f × f ) −1 s ϕ
(5.6)
= s ψ,
which completes the proof that ( f × f )−1 (υ) is a fuzzy TL-uniform base on X.
By a similar proof, we also have the following proposition.
K. A. Hashem and N. N. Morsi 23
Proposition 5.3. Let f j : X → Y j be a function, j ∈ J. If υ j is a fuzzy TL-uniform base on
Y j for every j ∈ J, then j ∈J ( f × f )−1 (υ j ) is a fuzzy TL-uniform base on X, where
j ∈J
(f × f)
−1
⎧
⎨0
υj = ⎩
(f × f)
−1
⎫
⎬
ϕ j : J1 is a finite subset of J, ϕ j ∈ υ j ∀ j ⎭ .
j ∈J1
(5.7)
Theorem 5.4. A source in TL-FUS ( f j : X → (Y j ,μ j )) j ∈J has an optimal lift in TL-FUS,
with fuzzy TL-uniform base
υ=
fj × fj
−1 μj .
j ∈J
(5.8)
Proof. Since ( f j × f j )−1 (μ j ) ⊆ j ∈J ( f j × f j )−1 (μ j ) = υ ⊆ υ∼T , then by Proposition 3.6, all
f j are uniformly continuous: (X,υ∼T ) → (Y j ,μ j ).
Let (Z, ℘) be in TL-FUS, and let a function h : Z → X be such that f j oh : (Z, ℘) →
(Y j ,μ j ) are uniformly continuous for all j ∈ J. Then by Proposition 3.6 again,
℘⊇
=
℘⊇
f j oh × f j oh
−1 f j × f j o(h × h)
j ∈J
(h × h)−1
⎡
= (h × h)−1 ⎣
μj
−1 fj × fj
μj
∀ j ∈ J, that is,
−1 fj × fj
μj
−1 j ∈J
⎤
(5.9)
μj ⎦
= (h × h)−1 (υ).
Referring to Proposition 3.6, once more, we find that h is uniformly continuous: (Z, ℘) →
(X,υ∼T ). This establishes that (X,υ∼T ) is the optimal lift in TL-FUS of the given source.
Proposition 5.5. TL-FUS is a topological category.
Definition 5.6. The functor -t∼ : TL-FUS → T-FLS leaves the functions unaltered, and is
defined on objects (X,μ) in TL-FUS by -t∼ (X,μ) = (X,τ(μ)).
Theorem 5.7. The above -t∼ is a well-defined functor. The proof follows immediately from
Theorem 3.10.
Theorem 5.8. The functor -t∼ preserves optimal lifts. The proof can be analogously as [17,
Theorem 7.3].
Theorem 5.9. Fuzzy TL-uniformizability is an initial property.
Proof. This follows from Proposition 5.5 and Theorem 5.8.
24
Fuzzy TL-uniform spaces
Acknowledgments
The authors express their sincere thanks to the referees and the Area Editor for their
thorough analysis of this manuscript and their most valuable suggestions which helped
improve this manuscript greatly.
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Khaled A. Hashem: Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt
E-mail address: khaledahashem@yahoo.com
Nehad N. Morsi: Basic & Applied Science Department, Arab Academy for Sciences & Technology
and Maritime Transport, P.O. Box 2033, Al-Horraya, Heliopolis, Cairo, Egypt
E-mail address: nehad@aast.edu